How can I store an animation action for later use?
Stash Them.
By stashing the actions we are doing little more than using the NLA as a bag to hold them, and associate them with their object.  If the object is linked or appended to another blend all the actions will come too. (Similarly with many export formats)  Pushing them down adds a track and action strip contributing to the animation result. Stashed actions do not.

Example of Action Editor above NLA Editor, with 3 previously stashed actions, Hop, Throb and Spin.  The new action &quot;Walk&quot; is in the slot, then stashed too.

To save an action it only needs to be stashed once.

Adding a new action to the slot in action editor automatically
stashes the prior.

The stashed animations do not contribute to the animation result.
(They are disabled by default), whereas those that are pushed down
do.

The top shown animation in the NLA editor, is the one in the slot..
(I call the active action the one in the slot, like putting a record
from your stash on the turntable. ... pushing down on the other hand
is like having multiple turntables, and some kind of mixing is
required. Would generally say if you are going to use NLA to animate
don't have one in the slot (unless editing)

Can select a stashed animation in the NLA and hit Tab to
put it in the &quot;slot&quot; to edit. (tweak) Shown as green in my colour scheme.

Have demonstrated editing by dragging keyframes in action editor, if
it's in the slot can also use timeline, dopesheet or inserting in 3d
view or over property field, or in graph editor to edit.